 Solution

In the language of a single edge relation $E$, axiomatize an undirected loop-free graph byFor every pair of nonnegative integers $r,s$, add the extension axiom demanding, for any distinct $u_1,\dots ,u_r,v_1,\dots ,v_s$, a new vertex $z$ such thatThese axioms form the theory of the random graph. Repetition of the fresh-vertex requirement also forces infinitude.

To prove quantifier elimination, first eliminate one existential quantifier from a quantifier-free formula. Put that formula into disjunctive normal form, so it suffices to consider a conjunction of literals in a new variable $z$ and the parameter variables. There are two possibilities for a witness: $z$ equals one of the parameters, in which case substitution gives a quantifier-free condition; or $z$ is distinct from all of them. In the latter case, the graph axioms reduce all requirements involving $z$ to specified adjacency or nonadjacency to the distinct parameter vertices. A requirement $E(z,z)$ is impossible, and conflicting requirements at equal parameter vertices are impossible. Every other consistent adjacency pattern is realized by the extension axiom.

There are only finitely many equality patterns of the given parameter tuple. For each, consistency of the literals is a quantifier-free condition, including the unchanged relations among the parameters. Their finite disjunction describes exactly when a witness exists. Thus $\exists z\theta (z,\mathbf{x})$ is equivalent to a quantifier-free formula whenever $\theta$ is quantifier-free. Mathematical induction on formulas now eliminates all quantifiers, using $\forall z\theta \equiv \neg \exists z\neg \theta$. Hence the random-graph theory eliminates quantifiers. The extension axioms also yield the usual countable random graph by a back-and-forth construction.